On a possible relation between Greek mathematics and Eleatic philosophy (Q6610496)

The subject of the paper under review is the relationship between Greek mathematics and Eleatic philosophy. This is a major subject in the history of Greek mathematics and mre generally in the history of human thought. Therefore, the reviewer thinks that a detailed analysis of some points of the author's thesis is deserved.\N\NThe author, in the paper under review, declares that his motivation was a work by Árpád Szabó written in the 1960s and translated into several languages [\textit{Á. Szabó}, Anfänge der griechischen Mathematik. München-Wien: R. Oldenbourg (1969; Zbl 0192.31603)]. The author shares some of Szabó's views and disagrees with some others. In particular, he is against the thesis that the Eleatics are at the origin of Greek mathematics (Szabó's thesis). On p. 177, he argues in favor of ``a reappraisal of Szabó's hypothesis about the origin of mathematics out of Eleatic philosophy. [\ldots] This interpretation restores the historically inverse relationship between mathematics and philosophy, refuting the attribution of mathematics' origin to a field outside mathematics, for which Szabó's hypothesis has been criticized.'' Regarding this central view of the paper under review, the reviewer thinks that the author is correct, but for the wrong reasons. The arguments will rely on recent papers by S. Negrepontis which will be quoted below. Let us be more specific.\N\NThe author, on p. 227, proposes that there is a basic similarity between Parmenides' semantic viewpoint and Pythagorean arithmetic, and that the Pythagoreans influenced Parmenides. He writes: ``all the propositions of Pythagorean arithmetic are affirmative'' (p. 226), that ``Parmenides' ontological universe, therefore, turns out to be a positive (negationless) true Being; there are no negative facts in it'' (p.225) and that ``Parmenides' theory of truth may have been reached through a process of reflexion on Pythagorean arithmetic in which truth was identified with genetic constructability'' (p. 227). Let us look closely at the author's arguments in support of his thesis:\N\N(1) He writes about Pythagorean arithmetic: ``Arithmetic is developed through genetic constructions in a variety of configurations'' (p. 219), and he talks about ``Genetic construction vs. proof.'' Thus, he says, ``the foundation of arithmetic in the neo-Pythagorean works is different in style from proof in the Euclidean Elements and the works of other mathematicians of classical antiquity. It amounts instead to a conception of effective genetic construction, using which the correctness of arithmetical statements is confirmed. This type of arithmetical reasoning about given numbers can be applied without assumptions of an axiomatic character. The confirmation of arithmetical statements is realizable through a specific kind of ``mental experiment'' (p. 221).\N\NHowever, Neo-Pythagoreans, such as Nicomachus and Theon, use indeed recursive rules starting from the unit to define arithmetical sequences; but these are recursive definitions and are not meant to replace rigorous proofs. One such ``genetic construction'' is the definition of the double sequence of the side and diameter numbers \(p_1=q_1=1, p_{n+1}=p_n+q_n, q_{n+1}=2p_n+q_n.\) These were defined after the discovery of the incommensurability of the diameter to the side of a square, and they are clearly its continued fraction convergents. The properties of this sequence are nevertheless rigorously proved by the Pythagoreans; for example the Pell property \(q_n^2=2p_n^2+(-1)^n\) is proved by means of the Pythagorean geometric proposition II.10 of the Elements. Cf. [Theon Smyrneus, ed.: Hiller 1878, 44,18--45,8, Iamblichus, Comments to Nikomachus, ed. Pistelli 1894, 92,23--93,6, and mainly Proclus, Commentary to Politeia, In Procli Diadochi, ed. Kroll 1899--1901 2, 24,16--25,13 and 2, 27, 1--29,4]. But the arithmetic of Book VII of the Elements, based on the Euclidean algorithm/anthyphairesis, is Pythagorean, clearly of an older period, since Philolaus in Fragment 6 already describes a musical anthyphairesis much more complicated than the arithmetical one (multiplicative and infinite). Book VII is based on the Principle of the Least (every strictly decreasing sequence of natural numbers is finite), used twice for the proof of the fundamental Propositions VII.2 and VII.31. It is not stated axplicitly as a Postulate in Book VII, as the geometrical Postulates in Book I of the Elements, but it is used in precisely the same way. The Principle of the Least is a most significant Axiom since, as we know today, it is equivalent to the Principle of mathematical Induction. This is important for the fact that the Pythagorean foundation of Arithmetic is closely related to Peano's axioms.\N\N(2) Vandoulakis writes about Parmenides: ``Parmenides' ontological universe, therefore, turns out to be a positive (negationless) true Being; there are no negative facts in it''. However, Plato in the Sophist 237a states Parmenides' objection to not-being. This holds for example in the following two passages:\N\NIn Sophist 237a, we read: ``(Stranger speaking) This statement involves the bold assumption that not-being exists, for otherwise falsehood could not come into existence. But the great Parmenides, my boy, from the time when we were children to the end of his life, always protested against this and constantly repeated both in prose and in verse: ``Never let this thought prevail, saith he, that not-being is; But keep your mind from this way of investigation'' (Parmenides Fr. 7). But eventually in 258c-d overcomes his objection, an overcoming that allows him in 262c to generate false statements.\N\NAnd in Sophist 258c-d, we read: ``(Stranger) Do you observe, then, that we have gone farther in our distrust of Parmenides than the limit set by his prohibition? (Theaetetus) What do you mean? (Stranger) We have proceeded farther in our investigation and have shown him more than that which he forbade us to examine. (Theaetetus) How so? (Stranger). Because he says somewhere: ``Never shall this thought prevail, that not-being is; Nay, keep your mind from this path of investigation.'' Parmenides Fr. 7.1 (Theaetetus) Yes, that is what he says. (Stranger) But we have not only pointed out that things which are not exist, but we have even shown what the form or class of not-being is.''\N\NThe recent analysis of Plato's Sophist and Parmenides by S. Negrepontis shows that Plato, in overcoming Parmenides' objection to not Being, imitates Zeno, and that Parmenides' objection amounts to his refusal to accept as true Being a philosophical imitation of the Pythagorean anthyphairetic dyad, proving the incommensurability of the diameter to the side of a square, which was generalized and streamlined by Theaetetus, and adopted by Plato as his model of true Being. In fact Zeno's true Being and Plato's intelligible Being essentially coincide; the dyad (One, Being) in the Parmenides is precisely the same as the dyad (Being, not-Being) in the Sophist, and they are both periodic anthyphairetic in nature. [\textit{B. Sriraman} (ed.), Handbook of the history and philosophy of mathematical practice. In 4 volumes. Cham: Springer. 599--644 (2024; Zbl 07819860); \textit{S. Negrepontis}, ``The Mysteries of Plato's Parmenides Dispersed. From musical intervals to contacts/``hapseis'' to ``logoi'''', in: Proceedings of the Conference ``Mathématiques et Musique, des Grecs a Euler'', (to appear)]\N\N(3) Vandoulakis writes about Zeno: ``In Scheme 1, we assume that proposition \(P\) is true, which leads us to absurdity. We then deduce that \(\neg P\) is true. In Scheme 2, we assume that proposition \(\neg P\) is true, which leads us to absurdity. We then conclude that \(\neg P\) is not true, i.e., \(\neg \neg P\) is true, and so that \(P\) is true. \dots Scheme 1 could have been used in such a case to reject incompatible (yet, not necessarily contradictory) alternatives (states). To cite one example, Zeno of Elea became famous for refuting an opponent's view with arguments analogous to those in Scheme 1.'' p. 228,\N\NHowever: In fact Zeno's basic argument (in Parmenides 127d6-128a3) is clearly as in Scheme 2. see [\textit{S. Negrepontis}, IRMA Lect. Math. Theor. Phys. 34, 949--998 (2023; Zbl 1534.00013)]: Step 1. Suppose the Many {=the sensibles} are/coincide with [true beings]. Step 2. The true beings are in Motion and at Rest (or infinite and finite, or one and many). Step 3. Hence, the Many are in Motion and at Rest (or infinite and finite, or one and many). Step 4. But this is clearly impossible. Step 5. Therefore the sensibles do not coincide with the true Beings.\N\NConcluding, while Vandoulakis is doubtlessly right in opposing Szabó's, and we might add [\textit{W. Burkert}, Weisheit und Wissenschaft: Studien zu Pythagoras. Philoloas und Platon (1962)] earlier, priority assignment to Eleatic Philosophy in relation with Pythagorean Mathematics, his arguments are open to objection. Pythagorean arithmetic is based on arithmetical anthyphairesis, which in no way excludes negation (for ex. Proof VII.2 and especially VII.31 of the Elements), (implicit) axioms (Principle of the least) and proofs, while Parmenides' objection to negation finds its precise explanation in his refusal to accept as model for the One, not the partless One, akin to the One of the first hypothesis, but the self-similar one, like the indivisible line, akin to the second hypothesis in the Parmenides. Incidentally, the partless One of the first hypothesis in the Parmenides, which would appear closer to Parmenides' concept of the true One, as it is only One and not One and Many as the One preferred by Zeno and Plato, is filled with negations.\N\NFor the entire collection see [Zbl 1544.03010].